એક \(T\) અર્ધઆયુવાળો રેડિયોએકિટવ ન્યુકિલયસ \(-A \) ન્યુકિલયસ \(-B\) માં ક્ષય પામે છે.\(t=0 \) સમયે ન્યુકિલયસ \(-A\) નથી.\(t -\) સમયે \(B\) ની સંખ્યા અને \(A\) ની સંખ્યાનો ગુણોત્તર \(0.3\) છે,તો \(t\) એ _______ વડે આપવામાં આવે :

A \(t \) = \(\frac{T}{2}\;\frac{{\log 2}}{{\log 1.3}}\)
B \(t = T\;\frac{{\log 1.3}}{{\log 2}}\)
C \(t=T log(1.3)\)
D \(t = \frac{T}{{{\rm{log}}\left( {1.3} \right)}}\;\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(t = T\;\frac{{\log 1.3}}{{\log 2}}\)

Step-by-step Solution

Detailed explanation

Let initially there are total \(\mathrm{N}_{0}\) number of nuclei At time \(\mathrm{t} \frac{N_{B}}{N_{A}}=0.3(\text { given })\) \(\Rightarrow \quad N_{B}=0.3 N_{A}\) \(\mathrm{N}_{0}=N_{A}+N_{B}=N_{A}+0.3 N_{A}\) \(\therefore \quad N_{A}=\frac{\mathrm{N}_{0}}{1.3}\) As we know…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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